Sierpinski-type fractals are differentiably trivial

Abstract

In this note we study generalized differentiability of functions on a class of fractals in Euclidean spaces. Such sets are not necessarily self-similar, but satisfy a weaker “scale-similar” property; in particular, they include the non self similar carpets introduced by Mackay–Tyson– Wildrick [12] but with different scale ratios. Specifically we identify certain geometric criteria for these fractals and, in the case that they have zero Lebesgue measure, we show that such fractals cannot support nonzero derivations in the sense of Weaver [16]. As a result (Theorem 26) such fractals cannot support Alberti representations and in particular, they cannot be Lipschitz differentiability spaces in the sense of Cheeger [3] and Keith [9]. 1. Motivation First order differentiable calculus has been extended from smooth manifolds to abstract metric spaces in many ways, by many authors. In this context, one important property of a metric space is the validity of Rademacher’s theorem, i.e. that Lipschitz functions are almost everywhere (a.e.) differentiable with respect to a choice of coordinates on that space. (For this reason, such spaces are known as Lipschitz differentiability spaces in the recent literature, e.g. [1, 2, 4] and said to have a measurable differentiable structure in earlier literature, e.g. [9, 11, 14].) The search for such a property naturally leads to questions of compatibility between a metric space and the choice of a Borel measure on that space. Even the case of Euclidean spaces has been addressed only recently. A result of De Phillipis and Rindler [5, Thm. 1.14] states that if Rademacher’s Theorem is true for a Radon measure μ on R, then μ must be absolutely continuous to m-dimensional Lebesgue measure. Here we address the case when μ is singular. As we will see, there is a large class of fractal sets, which we call Sierpiński-type fractals, for which Lipschitz functions do not even enjoy partial a.e. differentiability on the support of their natural measures. https://doi.org/10.5186/aasfm.2019.446